Isometry (Riemannian geometry)

In mathematics, an isometry of a manifold is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a metric on the manifold; a manifold with a (positive-definite) metric is a Riemannian manifold, one with an indefinite metric is a pseudo-Riemannian manifold. Thus, isometries are studied in Riemannian geometry.

A local isometry from one (pseudo-)Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first. When such a map is also a diffeomorphism, such a map is called an isometry (or isometric isomorphism), and provides a notion of isomorphism ("sameness") in the category Rm of Riemannian manifolds.

Definition

Let and be two (pseudo-)Riemannian manifolds, and let be a diffeomorphism. Then is called an isometry (or isometric isomorphism) if

where denotes the pullback of the rank (0, 2) metric tensor by . Equivalently, in terms of the push-forward , we have that for any two vector fields on (i.e. sections of the tangent bundle ),

If is a local diffeomorphism such that , then is called a local isometry.

See also

• Isometry group
• Myers–Steenrod theorem

References

• Lee, Jeffrey M. (2000). Differential Geometry, Analysis and Physics.